Scottish Higher Mathematics
Higher Mathematics
Higher Mathematics develops algebraic, geometric, trigonometric and calculus skills for Scottish Higher Mathematics. Use this area for topic support, worked examples, formula/reference help, mixed revision and teacher resources.
Topic library
Topic pages use Learn, Practise, Check and Answers sections with Higher Mathematics notation, short worked examples, generated practice and mixed revision for building exam-ready methods.
Course tools
Start with a topic page, use mixed revision for method choice, and keep formula support open for exact notation and reminders.
Algebra and Functions
Function notation, inverse and composite functions, domains, ranges, graph transformations and algebraic manipulation.
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Algebra and Functions
Function notation, inverse and composite functions, domains, ranges, graph transformations and algebraic manipulation.
Function notation and evaluation
Use f(x), substitute values, interpret outputs and connect functions to graphs.
Substitute accurately and state outputs clearly.
Composite and inverse functions
Build f(g(x)), reverse simple functions and handle domain restrictions for inverses.
Work in the correct order and check the inverse.
Domain, range and transformations
State sensible domains and ranges, then sketch translations, reflections and stretches.
Link notation to graph movement.
Straight Line and Circles
Coordinate geometry with gradients, line equations, distance, midpoint, circles and tangents.
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Straight Line and Circles
Coordinate geometry with gradients, line equations, distance, midpoint, circles and tangents.
Straight line methods
Find gradients, equations of lines and parallel or perpendicular relationships.
Use m = change in y / change in x.
Distance, midpoint and circles
Use coordinate formulae and interpret (x − a)² + (y − b)² = r².
Identify centre, radius and key points.
Tangents to circles
Use radius and tangent gradients to form line equations at a point on a circle.
Use perpendicular gradients.
Polynomials and Quadratics
Factor theorem, remainders, roots, discriminants, completing the square and graph features.
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Polynomials and Quadratics
Factor theorem, remainders, roots, discriminants, completing the square and graph features.
Factor and remainder theorem
Evaluate f(a), identify factors and use remainders to reason about polynomials.
Test roots by substitution.
Quadratics and discriminant
Solve quadratics, use the discriminant and connect algebraic form to graph behaviour.
Use b² - 4ac to classify roots.
Sketching polynomials
Use roots, intercepts, turning behaviour and sign information to sketch curves.
Mark roots and end behaviour first.
Differentiation
Power rule, gradients, tangents, normals, stationary points, optimisation and increasing or decreasing functions.
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Differentiation
Power rule, gradients, tangents, normals, stationary points, optimisation and increasing or decreasing functions.
Differentiating powers
Differentiate polynomial terms and evaluate gradients at given x-values.
Apply d/dx(xⁿ) = nxⁿ⁻¹.
Tangents, normals and rates of change
Find gradient functions, tangent equations, normal equations and rates of change.
Differentiate, substitute, then use line methods.
Stationary points and optimisation
Find stationary points, classify them and solve simple optimisation problems.
Set dy/dx = 0 and interpret the result.
Integration
Reverse differentiation, indefinite and definite integrals, area under curves and conditions.
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Integration
Reverse differentiation, indefinite and definite integrals, area under curves and conditions.
Indefinite integration
Reverse the power rule and include the constant of integration.
Increase the power and divide by the new power.
Definite integrals and area
Evaluate integrals between limits and interpret positive area under a curve.
Use F(b) - F(a).
Reverse derivatives with conditions
Find the original function from a derivative and a point or condition.
Integrate first, then use the condition to find C.
Trigonometry
Exact values, identities, equations, trig graphs and angle reasoning.
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Trigonometry
Exact values, identities, equations, trig graphs and angle reasoning.
Exact values and identities
Use exact trig values and identities such as sin² x + cos² x = 1.
Choose exact values before rounding.
Solving trigonometric equations
Solve equations over a given interval using symmetry and graph knowledge.
Find all solutions in the interval.
Trigonometric graphs
Interpret amplitude, period, phase shift and key features of trig graphs.
Read graph transformations carefully.
Exponentials and Logarithms
Exponential functions, logarithmic functions, log laws, equations and graph interpretation.
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Exponentials and Logarithms
Exponential functions, logarithmic functions, log laws, equations and graph interpretation.
Vectors
Vector notation, position vectors, magnitude, scalar product and geometry proofs.
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Vectors
Vector notation, position vectors, magnitude, scalar product and geometry proofs.
Vector operations and magnitude
Add, subtract and scale vectors, then calculate magnitudes from components.
Work component by component.
Position vectors and vector geometry
Use position vectors, routes and scalar multiples to prove geometric facts.
State the vector path clearly.
Scalar product
Use the scalar product to find angles and test perpendicular vectors.
Use a · b = |a||b|cos θ.
Recurrence Relations and Sequences
Generate terms, interpret recurrence relations and reason about limiting behaviour where appropriate.
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Recurrence Relations and Sequences
Generate terms, interpret recurrence relations and reason about limiting behaviour where appropriate.
Applications and Problem Solving
Multi-step Higher Mathematics problems requiring method choice, interpretation and clear reasoning.
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Applications and Problem Solving
Multi-step Higher Mathematics problems requiring method choice, interpretation and clear reasoning.